A parallel structured divide-and-conquer algorithm for symmetric tridiagonal eigenvalue problems

TL;DR

This paper introduces a parallel divide-and-conquer eigensolver for symmetric tridiagonal matrices that reduces communication and computation costs, demonstrating high scalability and improved performance over existing methods.

Contribution

The paper proposes a novel parallel structured divide-and-conquer algorithm utilizing Cauchy-like matrix properties and low-rank approximations for efficient eigenvalue computations.

Findings

Scales efficiently to at least 4096 processes.

Outperforms PHDC with better scalability.

Faster than ScaLAPACK's PDSTEDC and comparable to ELPA.

Abstract

In this paper, a parallel structured divide-and-conquer (PSDC) eigensolver is proposed for symmetric tridiagonal matrices based on ScaLAPACK and a parallel structured matrix multiplication algorithm, called PSMMA. Computing the eigenvectors via matrix-matrix multiplications is the most computationally expensive part of the divide-and-conquer algorithm, and one of the matrices involved in such multiplications is a rank-structured Cauchy-like matrix. By exploiting this particular property, PSMMA constructs the local matrices by using generators of Cauchy-like matrices without any communication, and further reduces the computation costs by using a structured low-rank approximation algorithm. Thus, both the communication and computation costs are reduced. Experimental results show that both PSMMA and PSDC are highly scalable and scale to 4096 processes at least. PSDC has better scalability than PHDC that was proposed in [J. Comput. Appl. Math. 344 (2018) 512--520] and only scaled to 300 processes for the same matrices. Comparing with \texttt{PDSTEDC} in ScaLAPACK, PSDC is always faster and achieves $1.4$x--$1.6$x speedup for some matrices with few deflations. PSDC is also comparable with ELPA, with PSDC being faster than ELPA when using few processes and a little slower when using many processes.